MITSUBISHI ELECTRIC RESEARCH LABORATORIES http://www.mer.com Effect of Random Wak Phase Noise on MIMO Measurements Peter Amers, Shurjee Wyne, Fredrik Tufvesson and Andreas Moisch TR25-59 Juy 25 Abstract In this paper we study the infuence of phase noise from free-running oca osciators on SAGE signa parameter estimation. Phase noise is here modeed as a random-wak process. We present phase noise estimates from our LUND RUSK MIMO channe sounder, and draw concusions on requirements on oca osciators phase noise in terms of the Aan variance. We investigate an error propagation effect in SAGE, and finay, we present random-wak phase noise effect on channe capacity. IEEE Vehicuar Technoogy Conference, 25 This work may not be copied or reproduced in whoe or in part for any commercia purpose. Permission to copy in whoe or in part without payment of fee is granted for nonprofit educationa and research purposes provided that a such whoe or partia copies incude the foowing: a notice that such copying is by permission of Mitsubishi Eectric Research Laboratories, Inc.; an acknowedgment of the authors and individua contributions to the work; and a appicabe portions of the copyright notice. Copying, reproduction, or repubishing for any other purpose sha require a icense with payment of fee to Mitsubishi Eectric Research Laboratories, Inc. A rights reserved. Copyright c Mitsubishi Eectric Research Laboratories, Inc., 25 21 Broadway, Cambridge, Massachusetts 2139
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Effect of Random Wak Phase Noise on MIMO Measurements Peter Amers 1,2, Shurjee Wyne 1, Fredrik Tufvesson 1 and Andreas F. Moisch 1,3 1 Dept. of Eectroscience, Lund University, Box 118, SE-221 Lund, Sweden. 2 TeiaSonera AB, Box 94, SE-21 2 Mamö, Sweden. 3 Mitsubishi Eectric Research Labs, Cambridge, MA 2139, USA. Emai: {Peter.Amers, Shurjee.Wyne, Fredrik.Tufvesson, Andreas.Moisch}@es.th.se Abstract In this paper we study the infuence of phase noise from free-running oca osciators on SAGE signa parameter estimation. Phase noise is here modeed as a random-wak process. We present phase noise estimates from our LUND RUSK MIMO channe sounder, and draw concusions on requirements on oca osciators phase noise in terms of the Aan variance. We investigate an error propagation effect in SAGE, and finay, we present random-wak phase noise effect on channe capacity. I. INTRODUCTION Mutipe-input mutipe-output (MIMO) wireess communication systems have muti-eement antenna arrays at both the transmitter and the receiver side. It has been shown that they have the potentia for arge information-theoretic capacities since the system can provide severa independent communications channes between transmitter and receiver [1], [2], [3]. Measurements of the spatia radio channe [4] are vita for understanding the MIMO channe and for system design, simuations and performance anaysis. The most popuar method for obtaining such measurements is by a channe sounder with a switched array, where the eements of an antenna array are connected, one after the other, to a conventiona channe sounder measuring impuse responses [5]. However, MIMO measurements obtained by either this, or any other, principe suffer from severa error sources in addition to therma additive white Gaussian noise (discussed in [6], [7]), e.g. phase noise (PN) during the measurements and array caibration errors. PN, i.e. frequency fuctuations of the oca osciators (LOs) in the transmitter and receiver, can be divided into two main categories: (i) when the system is phase-ocked (e.g. transmitter and receiver are connected and use the same LO) the resuting PN is ow and modeed as a zero mean, stationary, finite-power random process; (ii) when the system is frequency-ocked ony, often termed free-running, the resuting PN is sowy varying but not imited, and is modeed as a zero-mean non-stationary infinite power Wiener process, e.g., a random-wak process. Whie phase-ocked measurements can be performed in indoor environments, micro- and macroceuar environments impy a arge separation between transmitter and receiver, and thus dictate the use of freerunning osciators. Severa authors have investigated the impact of PN on mutiantenna systems. Kivinen et a. [8] investigated the impact of free-running osciator PN effect on Fourier-based direction of arriva (DOA) estimation for singe-input mutipe-output (SIMO) systems. A recent contribution, [9], investigates the impact of PN in phase-ocked LOs on the capacity of fuy correated rank-one channes as we as fu rank channes. This paper focuses on the impact of PN from free-running osciators in MIMO systems, investigating its impact on estimated anges, deay, and Dopper of the mutipath components (MPCs), as we as its effect on the estimated channe capacity. In contrast to [8], we study the impact of the PN on the MPC parameters obtained by the high-resoution SAGE agorithm [1], which has become the de-facto standard for the evauation of doube-directiona measurement campaigns. In contrast to [9], we investigate the impact of free-running osciators on the capacity. We aso present PN measurement on our RUSK LUND channe sounder and discuss the error propagation effect in the SAGE agorithm. The remainder of the paper is organized the foowing way: Sec. II describes the modes and theoretica background for both the SAGE agorithm and the channe capacity, as we as measurements of the PN in our channe sounder. Section III presents the simuation resuts for the impact of PN on the estimated directions of the MPCs and the capacity. A summary and concusions wrap up this paper. II. THEORY AND MODELS A. Osciator PN Mode and Aan Variance Assume that we have a LO generator with an instantaneous output votage V (t) =V sin (2πf c t + ϕ (t)), (1) where V and f c are the nomina ampitude and carrier frequency, respectivey. The time variations of the frequency are incorporates in the phase noise ϕ (t) and the actua (instantaneous) frequency becomes f (t) =f c + d ϕ (t) dt 2π. (2) The phase noise is the difference between phase of the carrier and the phase of the LO, and the fractiona frequency deviation is defined as [11], [12] y (t) = f (t) f c f c = d ϕ (t). (3) dt 2πf c -783-8887-9/5/$2. (c)25 IEEE
If the system is ony frequency ocked but not phase ocked, the LO is said to be free-running. The instantaneous frequency of a free-running LO is we characterized by a fractiona frequency deviation, y (t), modeed as a zero mean white Gaussian random process. Since ϕ (t) is obtained by integrating the frequency deviation y (t), the assumptions on y (t) impy that the phase noise ϕ (t) can be described as a random wak process (continuous-path Brownian motion or Wiener Lévy process) with zero mean and a variance that increases ineary with time. The time-average of the fractiona frequency deviation is ȳ k (t) = 1 τ tk +τ t k y (t) dt = ϕ (t k + τ) ϕ (t k ), (4) 2πf c τ where τ is the averaging period, here equa to the period between two successive measurements, and t k+1 = t k + τ. Another important quantity for the characterization of osciators is the Aan variance, which is defined as [11], [12] σy 2 (ȳ k+1 ȳ k ) 2 (τ) =, (5) 2 1 5 Individua reference 5 1 2 3 4 5 15 Phase [deg] 1 5 1 2 3 4 5 15 16 17 18.5.1.15 Time [s] 1 2 Common reference 3 1 2 3 4 5 1 1 2 1 2 3 4 5 1 1 2.5.1.15 Time [s] Fig. 1. Phase noise measured from the RUSK LUND MIMO channe sounder, with (a) free running LOs at transmitter and revceiver, and (b) for a common reference. 1 8 1 9 where defines the infinite-time average. It can be estimated a finite data set as [11], [12] (τ) 1 1 ˆσ 2 y (τ) = 1 K = 1 K K (ȳ k+1 ȳ k ) 2 2 K (ϕ (t k+2 ) 2ϕ (t k+1 )+ϕ(t k )) 2 2(2πf c τ) 2. (6) k=1 k=1 In Fig. 1 (a) we present resuts from back-to-back measurement of the PN in the RUSK LUND MIMO channe sounder. The measurement was performed with a center frequency of 5.4 GHz and a bandwidth of 24 MHz. For the eft subpots, the transmit and receive LO were free-running and the pots show the sum of the PN in the transmitter and receiver. In the right subpots, the same LO was used for the transmitter and receiver. The random wak behavior is evident in the upper eft subpot. The two LOs are not measured individuay, and other sources of noise are present. For shorter time periods, these noise sources dominate and hide the effect of the time dependent random-wak PN, see e.g. second and third subpot to the eft. In the paper we ony consider the effect of randomwak PN, the effect of PN might be even more pronounced for other PN modes. The Aan variance corresponding to the measurements in Fig.1 for the free-running configuration is shown in Fig. 2. Note that the Aan variance is cacuated as the sum of the PN in the transmitter and receiver. The Aan variance of the system (sum of transmitter and receiver PN) at 1 s, (1s) 7 1 12. B. MIMO Signa Mode In order to obtain a MIMO channe mode and extract the parameters of the MPCs, we use the SAGE agorithm [1]. Fig. 2. 1 11 1 12 1 6 1 5 1 4 1 3 1 2 1 1 1 τ [s] The (1s) for the RUSK LUND MIMO channe sounder. In a first step, the SAGE requires to mode the impact of an arbitrary MPC on the received signa s : ( ) ( ) s (k, i; θ )=α e j2π( tν i f τ k) c r φ (r) c T t, (7) φ (t) where i, is the snapshot number, k is the frequency index, the index of the considered MPC, and θ the set of parameters describing the MPC; this set incudes α, the compex ampitude, ν, the Dopper frequency, τ, is the deay and φ (r), φ (r) is the direction of arriva (DOA) and direction of departure (DOD). Furthermore, t is the time between snapshots, f is the spacing between measured frequencies, and c r and c t are the steering vectors at receiver and transmitter, respectivey. For each of the N t transmit eements, we assume that the N r receive eements at the receiver are swept with an eement switching time of T mux. Hence, the switching time between transmitter array eements is then N r T mux, and the time between two consecutive snapshots is N r N t T mux. We assume that the random-wak PN affects a frequency sub-channes of the muticarrier spread spectrum test signa equay, due to the short time duration of the test signa. Thus the PN corrupted
signa is modeed as s PN (k, i; θ )=s (k, i; θ ) N i, (8) where is the Schur product, and N i = e jϕ1,i e jϕnr+1,i e jϕ (Nr(Nt 1)+1),i e jϕ2,i e jϕnr+2,i e jϕ (Nr(Nt 1)+2),i......, (9) e jϕnr,i e jϕ2nr,i e jϕnrn t,i where ϕ q,i = ϕ (qt mux +(i 1) N r N t T snap ) is the randomwak phase error kt mux +(i 1) N r N t T snap seconds after caibration. C. MIMO Channe Mode and Capacity To investigate the PN effect on channe capacity evauations, the error in estimated capacity depends on the actua channe transfer matrix, H, in the input-output reation r = Hx + n, (1) where r is the received coumn vector, x is the transmit coumn vector and n is the additive white Gaussian noise coumn vector. We focus on the worst-case scenario, that is the one that minimizes the channe capacity. This occurs when the channe transfer matrix has a singe eigenmode, i.e., ony one nonzero eigenvaue. A channe matrix that satisfies this condition is the keyhoe channe [13], [14] defined as H key = g R g T, (11) where g R, g T are) coumn vectors distributed as g R, g T CN (,σkey 2 I, thus the entries are uncorreated [13]. The transfer matrix corrupted by PN is then defined as H PN = H key N i. (12) The instantaneous capacity is [2], [3] ) C = og 2 det (I max(nr,nt) + γnt W H, (13) where { HH W H = H H for N r N t for N r >N t. (14) and I max(nr,n t) is an identity matrix of size max (N r,n t ) max (N r,n t ). The resuts of the capacity evauations are presented in the next section. III. SIMULATION RESULTS To anayze the infuence of the PN on the MPC parameters as estimated by SAGE, we study two scenarios: (i) a synthetic scenario with a singe MPC, and (ii) a reaistic scenario with 1 MPCs. The parameters used for the simuations are presented in Tabe I. As a criterion of the error, we use the root mean square error (RMSE) for the DOA and DOD, defined as ε RMSE = x ˆx n 2, (15) n ε RMSE [deg] 5 45 4 35 3 25 2 15 1 5 TABLE I PARAMETERS USED FOR SIMULATIONS 1 1 1 9 1 8 (1s) SAGE parameters Vaue No iterations (for RMSE) 1 No SAGE iterations 4 No snapshots 2 T mux 5 µs φ (r) 1,φ(t) 1 (for singe MPC) ULA: DOD ULA: DOA UCA: DOD UCA: DOA Rea ULA: DOD Rea ULA: DOA ε re 2 1.8 1.6 1.4 1.2 1.8.6.4.2 ULA: Ampitude ULA: Deay UCA: Ampitude UCA: Deay Rea ULA: Ampitude Rea ULA: Deay 1 1 1 9 1 8 (1s) ε re 15 1 5 ULA: Dopper UCA: Dopper Rea ULA: Dopper 1 1 1 9 1 8 (1s) Fig. 3. RMSE versus (1s) for an 8 8 ULA system, an 8 8 UCA system both with uniform eement responses, and, finay, RMSE for a rea patch ULA antenna. The MPC departure and arrives at broad side of the ULAs. where n is the iteration number. For deay, ampitude and Dopper frequency the reative error, ε re, is used as error measure, defined as ε re = 1 n 1 n x ˆx n. (16) x A. Singe MPC In Fig. 3 the effect of random-wak PN on estimated DOA, DOD, ampitude, deay and Dopper frequency is presented for a singe MPC when using a 8 8 systems with: two uniform inear arrays (ULAs), two uniform circuar arrays (ULAs) (ULA and UCA, both with idea uniform antenna patterns) and finay, a 8 8 system with two rea patch ULAs (i.e. with directiona antenna patterns, see [15] for detais). The DOD is more affected by random-wak PN than the DOA, due to the N r -times onger eement switching time between transmitter eements. The UCA seems to be more sensitive to random-wak PN than the inear arrays. The directivity of the rea array eement makes the broadside MPC more affected by random-wak PN, however the ampitude error is simiar for a configurations and is a resut of mismatch between signa mode and the actua signa, incuding PN. The time between sampes for Dopper estimation is (N r N t T mux ), thus due to the reative ong time between sampe of the Dopper frequency its estimates are more effected by the time dependent random-wak PN. The ULA and UCA has a worse Dopper frequency estimate than the rea ULA. In Fig. 4 the effect of array size on the DOA and DOD errors is shown, at an Aan variance of (1s) =1 8.The
25 2 ULA: DOD ULA: DOA UCA: DOD UCA: DOA 1 ε RMSE [deg] 15 1 Power [dbm] 2 3 4 5 2 4 8 16 No. eements Fig. 4. RMSE versus number of eements of the ULA and UCA arrays at an Aan variance of (1s) =1 8. Each array consists of eements with uniform eement responses. 5 Power of existing MPC Power of ghost MPC 6 1 12 1 11 1 1 1 9 1 8 1 7 σ (1s) y Fig. 5. The power of the existing MPC with a power of dbm and the power of the ghost component as a function of Aan variance. RMSE of the UCA for DOD increases with the array size, whereas for the DOA the RMSE is amost independent of the array size. For ULA both DOD and DOA appears to be independent of the array size. 3 4 5 DOD (1s)=1 12 3 σy (1s)=1 12 4 5 DOA B. Error Propagation When there is a mismatch of the signa mode and the measured signa (incuding PN), the residua of the canceed MPCs might be interpreted by the agorithm as additiona MPCs, here denoted ghost components, which physicay are non-existing. An estimated ghost component wi be used in the next SAGE iteration, hence the error wi propagate. The error propagation due to the noise estimation step in the SAGE agorithm [1] is given by y (t) L =1 ( s t; ˆθ ), (17) ( where y (t) is the measured signa (MIMO channe), s t; ˆθ ) is the continuous SAGE signa mode [1], and ˆθ is the parameter vector for the first estimate (( ) ) and the th MPC. In Fig. 5 the power of the rea MPC and the strongest ghost component is potted versus the Aan variance. If the ghost component becomes strong enough, it wi beong to the set of estimated MPCs. In addition to this, the mismatch of the signa mode to the rea signa resuts in a worse correation and therefore a ower estimated ampitude of the rea MPC. Thus, with a measurement dynamic range of 3 db, the Aan variance has to be ess than (1s) =1 1 to suppress the ghost component and avoid the ower correation gain. A simiar behavior wi be present for other signa mode mismatches, e.g., when the pane wave assumption does not hod, when the sma bandwidth assumption is vioated, or due to array caibration errors. Power [dbm] 3 4 5 3 4 (1s)=1 11 (1s)=1 1 5 2 1 1 Ange [deg] 3 σy (1s)=1 11 4 5 3 σy (1s)=1 1 4 5 1 5 5 Ange [deg] Fig. 6. The DOD, DOA and power estimates (marked with ) for1 MPC generated with the SAGE signa mode, for parameters taken from a rea LOS measurement scenario (marked with ). The 1 SAGE estimates for each of the 1 channe reaizations are marked with ( ). C. Mutipe MPCs Here we evauate the effect of random-wak PN on estimated SAGE parameters with mutipe MPCs, for two scenarios: ineof-sight (LOS) and non-los. In order to consider reaistic vaues, we use parameter vaues of 1 MPCs from a measurement campaign described in [15]. The parameters used for the 1 MPCs are marked with ( ) in Fig. 6 and Fig. 7, for a LOS and a non-los scenario, respectivey. For each Aan variance, 1 iterations are made and the SAGE estimates are marked with ( ). In the figures it can be seen that there are some MPCs that are not detected by SAGE and some MPCs that are ghost components. Here the difference between the strongest and weakest MPCs is ess than 2 db. With a arger dynamic range the effect of error propagation and ghost components wi be more pronounced.
Power [dbm] 5 σy (1s)=1 12 6 7 8 5 σy (1s)=1 11 6 7 8 5 σy (1s)=1 1 6 7 DOD 8 4 3 2 1 Ange [deg] 5 σy (1s)=1 12 6 7 8 5 σy (1s)=1 11 6 7 8 5 σy (1s)=1 1 6 7 DOA 8 8 4 4 8 Ange [deg] Fig. 7. The DOD, DOA and power estimates (marked with ) for1 MPC generated with the SAGE signa mode, for parameters taken from a rea non-los measurement scenario (marked with ). The 1 SAGE estimates for each of the 1 channe reaizations are marked with ( ). Average capacity [bit/s/hz] 8 7 6 5 4 3 2 1 Ckey @ SNR = 2 db C with PN 16x16 C with PN 8x8 C with PN 4x4 C with PN 2x2 1 11 1 1 1 9 1 8 1 7 1 6 σ (1s) y Fig. 8. The average channe capacity versus Aan variance (square root), for different system sizes. D. PN Effect on Channe Capacity In Fig. 8 cacuations of the average capacity for four different antenna configurations are potted versus the Aan variance for an evauation SNR of 2 db. From the figure it can be concuded that, for this evauation SNR, the Aan variance of the measurement equipment has to be ess than (1s) < 1 1 to give capacity vaues cose to the idea case. Note that due to the properties of the keyhoe channe, the impact of the PN is maximum. Bad LOs can thus contribute to the probems of measuring rea-ife keyhoe channes. IV. SUMMARY In this paper we have studied the effect of random-wak PN (in terms of Aan variance of the sum of the PN of two freerunning LOs) on the SAGE estimation agorithm. We have study two MIMO scenarios: (i) a synthetic scenario with a singe MPC, and (ii) two reaistic scenario with 1 MPCs (LOS and NLOS). We discuss an error propagation effect inherent in SAGE and study the resuting ghost components versus the Aan variance. Finay, we present the randomwak PN effect on capacity estimates. From our simuations it seems that the Aan variance of the osciators shoud be in the order of (1s) 1 1, for an eement switching time of T mux =5µs. As a comparison the LUND RUSK MIMO channe sounder has an Aan variance of (1s) 1 11. Note that for a onger switching time, the random-wak PN effect wi be more pronounced and the requirements on the osciators become higher. E.g. a doubing of the switch time has the same effect as doubing the Aan variance. Acknowedgement 1: The authors woud ike to thank Gunnar Eriksson for his hep with the SAGE agorithm. This work was party funded by an INGVAR grant of the Swedish Foundation for Strategic Research and a grant from Vetenskapsrådet. REFERENCES [1] J. H. Winters, On the capacity of radio communications systems with diversity in Rayeigh fading environments, IEEE Journa on Seected Areas in Communications, vo. 5, pp. 871 878, June 1987. [2] G. J. Foschini and M. J. Gans, On imits of wireess communications in fading environments when using mutipe antennas, Wireess Persona Communications, vo. 6, pp. 311 335, 1998. [3] I. E. 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